(i) How many symmetric matrices, of order 3 can be formed using the first ten single digit whole

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(i) How many symmetric matrices, of order 3 can be formed...

Key Concepts

Symmetric Matrix

A symmetric matrix is one in which the element in the i-th row and j-th column is equal to the element in the j-th row and i-th column. This characteristic means that when forming such a matrix, the entries above (or below) the main diagonal determine those below (or above) the diagonal, reducing the number of independent elements. This concept is important when considering structural properties and counting the number of distinct matrices that can be formed under symmetry constraints.

Skew Symmetric Matrix

A skew symmetric matrix is defined by the property that its transpose is equal to the negative of the matrix, which implies that each diagonal element must be zero, and every off-diagonal pair of elements must be opposites of each other. The unique structure of skew symmetric matrices plays a significant role in problems where counting such matrices involves acknowledging that only certain elements (typically in the upper or lower triangular part, excluding the diagonal) can be chosen independently while the others are automatically determined by them.

Combinatorial Counting in Matrix Structures

Combinatorial counting in the context of matrix construction involves determining the number of ways to fill the entries of a matrix subject to specified rules and constraints, such as symmetry or skew symmetry. Important principles include identifying independent entries, applying the multiplication rule for choices from a set of numbers, and taking into account restrictions imposed either by the available elements or structural properties of the matrix.

Matrix Order and Entry Constraints

The order of a matrix refers to its size (number of rows and columns) and directly influences the number of entries that need consideration. In counting problems, the matrix order, combined with specific constraints (like those for symmetric and skew symmetric matrices), determines which entries can be chosen arbitrarily and which are dependent, ultimately affecting the total count of possible matrices. Understanding these constraints is vital for setting up and solving enumeration problems in matrix theory.

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